```python
from typing import List, Tuple

def split_agenda(sessions: List[Tuple[int, int]], limit: int) -> List[Tuple[int, int]]:
    if not sessions or not limit:
        raise ValueError("Sessions or limit cannot be empty.")
    
    if not all(isinstance(session, tuple) and len(session) == 2 for session in sessions):
        raise ValueError("Sessions must be a list of tuples.")
    
    if not all(isinstance(start, int) and isinstance(end, int) for start, end in sessions):
        raise ValueError("Session start and end times must be integers.")
    
    if not all(start < end for start, end in sessions):
        raise ValueError("Sessions must be in chronological order.")
    
    if not all(isinstance(limit, int) and limit > 0):
        raise ValueError("Limit must be a positive integer.")
    
    blocks = []
    for session in sessions:
        start, end = session
        if end - start > limit:
            mid = (start + end) // 2
            if (end - mid) % 2 == 1:
                mid += 1
            blocks.append((start, mid))
            blocks.append((mid, end))
        else:
            blocks.append(session)
    
    return blocks
```